Additive sumset sizes with tetrahedral differences

TL;DR

This paper proves that for 4-element integer sets, the sizes of their h-fold sumsets are always among specific numbers related to tetrahedral differences, confirming experimental observations with explicit constructions.

Contribution

It establishes the existence of these 'popular' sumset sizes and provides explicit constructions for each, advancing understanding of sumset size distributions.

Findings

Sumset sizes are concentrated at tetrahedral difference numbers.

Explicit constructions for each sumset size are provided.

Theoretical proof confirms experimental observations.

Abstract

Experimental calculations suggest that the $h$-fold sumset sizes of 4-element sets of integers are concentrated at $h$ numbers that are differences of tetrahedral numbers. In this paper it is proved that these "popular" sumset sizes always exist. Explicit $h$-adically defined sets are constructed for each of these numbers.